Place the beans, broth, and lime juice in a large bowl.,Use a potato masher to mash the mixture into a course puree.,Stir in the cilantro, chile, chili powder, and salt and pepper to taste.,Stir in 1/4 cup cornmeal.,______,Shape the mixture into eight 3-inch cakes.,Heat 1 1/2 tablespoons of the oil in a large nonstick skillet over medium heat until shimmering.,Add the bean cakes and cook until nicely browned, about 4 minutes.,Carefully turn the cakes and drizzle the remaining 1 1/2 tablespoons oil around the edges of the pan.,Continue cooking until the bean cakes are nicely browned on the se4cond side, about 3 minutes.,Serve immediately.